| L(s) = 1 | + (−1.55 + 1.25i)2-s + (0.862 − 3.90i)4-s + (0.693 − 1.20i)5-s + (−0.562 − 0.975i)7-s + (3.54 + 7.17i)8-s + (0.422 + 2.73i)10-s + (−1.11 − 1.92i)11-s + (−14.4 − 8.34i)13-s + (2.09 + 0.815i)14-s + (−14.5 − 6.73i)16-s − 20.2i·17-s − 21.0i·19-s + (−4.09 − 3.74i)20-s + (4.14 + 1.60i)22-s + (18.0 + 10.4i)23-s + ⋯ |
| L(s) = 1 | + (−0.779 + 0.626i)2-s + (0.215 − 0.976i)4-s + (0.138 − 0.240i)5-s + (−0.0804 − 0.139i)7-s + (0.443 + 0.896i)8-s + (0.0422 + 0.273i)10-s + (−0.101 − 0.175i)11-s + (−1.11 − 0.642i)13-s + (0.149 + 0.0582i)14-s + (−0.907 − 0.421i)16-s − 1.19i·17-s − 1.10i·19-s + (−0.204 − 0.187i)20-s + (0.188 + 0.0731i)22-s + (0.784 + 0.452i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.369 + 0.929i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.369 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.637224 - 0.432525i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.637224 - 0.432525i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.55 - 1.25i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-0.693 + 1.20i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (0.562 + 0.975i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (1.11 + 1.92i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (14.4 + 8.34i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 20.2iT - 289T^{2} \) |
| 19 | \( 1 + 21.0iT - 361T^{2} \) |
| 23 | \( 1 + (-18.0 - 10.4i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (26.9 + 46.7i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-9.19 + 15.9i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 34.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-15.1 - 8.72i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-44.2 + 25.5i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (32.4 - 18.7i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 52.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + (27.1 - 46.9i)T + (-1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (76.5 - 44.2i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-66.0 - 38.1i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 69.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 82.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-19.8 - 34.3i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-56.9 - 98.6i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 25.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-10.4 - 18.0i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.64595385092865837842129854388, −10.80721350102275784622041976066, −9.597917222065318910218117401693, −9.143561327617704799139777903182, −7.71295463434325179220619831582, −7.12863640595846941915069019317, −5.70894742346127514858331145692, −4.79169638968846103446951508990, −2.58262215464860575901751198307, −0.54520095880555069153512469266,
1.75198794053997514572572808541, 3.14848416357518825506492637497, 4.60783071119669201595540928534, 6.37177109513339514260242193534, 7.41588532973555997983122540255, 8.473025894246687565339832114474, 9.452933699789313365217991709231, 10.34275621929629450802900729660, 11.08659019006539292288292080713, 12.41855263603152124260503442750
